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Michael K. J. Goodman An Introduction to the Early Development of Mathematics

An easy-to-read presentation of the early history of mathematics Engaging and accessible, An Introduction to the Early Development of Mathematics provides a captivating introduction to the history of ancient mathematics in early civilizations for a nontechnical audience. Written with practical applications in a variety of areas, the book utilizes the historical context of mathematics as a pedagogical tool to assist readers working through mathematical and historical topics. The book is divided into sections on significant early civilizations including Egypt, Babylonia, China, Greece, India, and the Islamic world. Beginning each chapter with a general historical overview of the civilized area, the author highlights the civilization’s mathematical techniques, number representations, accomplishments, challenges, and contributions to the mathematical world. Thoroughly class-tested, An Introduction to the Early Development of Mathematics features: Challenging exercises that lead readers to a deeper understanding of mathematics Numerous relevant examples and problem sets with detailed explanations of the processes and solutions at the end of each chapter Additional references on specific topics and keywords from history, archeology, religion, culture, and mathematics Examples of practical applications with step-by-step explanations of the mathematical concepts and equations through the lens of early mathematical problems A companion website that includes additional exercises An Introduction to the Early Development of Mathematics is an ideal textbook for undergraduate courses on the history of mathematics and a supplement for elementary and secondary education majors. The book is also an appropriate reference for professional and trade audiences interested in the history of mathematics. Michael K. J. Goodman is Adjunct Mathematics Instructor at Westchester Community College, where he teaches courses in the history of mathematics, contemporary mathematics, and algebra. He is also the owner and operator of The Learning Miracle, LLC, which provides academic tutoring and test preparation for both college and high school students.

5245.15 рублей

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Richard Henriksen N. Scale Invariance. Self-Similarity of the Physical World

Bringing the concepts of dimensional analysis, self-similarity, and fractal dimensions together in a logical and self-contained manner, this book reveals the close links between modern theoretical physics and applied mathematics. The author focuses on the classic applications of self-similar solutions within astrophysical systems, with some general theory of self-similar solutions, so as to provide a framework for researchers to apply the principles across all scientific disciplines. He discusses recent advances in theoretical techniques of scaling while presenting a uniform technique that encompasses these developments, as well as applications to almost any branch of quantitative science. The result is an invaluable reference for active scientists, featuring examples of dimensions and scaling in condensed matter physics, astrophysics, fluid mechanics, and general relativity, as well as in mathematics and engineering.

2095.78 рублей

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Donald Bindner Mathematics for the Liberal Arts

Presents a clear bridge between mathematics and the liberal arts Mathematics for the Liberal Arts provides a comprehensible and precise introduction to modern mathematics intertwined with the history of mathematical discoveries. The book discusses mathematical ideas in the context of the unfolding story of human thought and highlights the application of mathematics in everyday life. Divided into two parts, Mathematics for the Liberal Arts first traces the history of mathematics from the ancient world to the Middle Ages, then moves on to the Renaissance and finishes with the development of modern mathematics. In the second part, the book explores major topics of calculus and number theory, including problem-solving techniques and real-world applications. This book emphasizes learning through doing, presents a practical approach, and features: A detailed explanation of why mathematical principles are true and how the mathematical processes work Numerous figures and diagrams as well as hundreds of worked examples and exercises, aiding readers to further visualize the presented concepts Various real-world practical applications of mathematics, including error-correcting codes and the space shuttle program Vignette biographies of renowned mathematicians Appendices with solutions to selected exercises and suggestions for further reading Mathematics for the Liberal Arts is an excellent introduction to the history and concepts of mathematics for undergraduate liberal arts students and readers in non-scientific fields wishing to gain a better understanding of mathematics and mathematical problem-solving skills.

7423.46 рублей

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Roger L. Cooke The History of Mathematics. A Brief Course

Praise for the Second Edition «An amazing assemblage of worldwide contributions in mathematics and, in addition to use as a course book, a valuable resource . . . essential.» —CHOICE This Third Edition of The History of Mathematics examines the elementary arithmetic, geometry, and algebra of numerous cultures, tracing their usage from Mesopotamia, Egypt, Greece, India, China, and Japan all the way to Europe during the Medieval and Renaissance periods where calculus was developed. Aimed primarily at undergraduate students studying the history of mathematics for science, engineering, and secondary education, the book focuses on three main ideas: the facts of who, what, when, and where major advances in mathematics took place; the type of mathematics involved at the time; and the integration of this information into a coherent picture of the development of mathematics. In addition, the book features carefully designed problems that guide readers to a fuller understanding of the relevant mathematics and its social and historical context. Chapter-end exercises, numerous photographs, and a listing of related websites are also included for readers who wish to pursue a specialized topic in more depth. Additional features of The History of Mathematics, Third Edition include: Material arranged in a chronological and cultural context Specific parts of the history of mathematics presented as individual lessons New and revised exercises ranging between technical, factual, and integrative Individual PowerPoint presentations for each chapter and a bank of homework and test questions (in addition to the exercises in the book) An emphasis on geography, culture, and mathematics In addition to being an ideal coursebook for undergraduate students, the book also serves as a fascinating reference for mathematically inclined individuals who are interested in learning about the history of mathematics.

10272.62 рублей

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Theodore Faticoni G. The Mathematics of Infinity. A Guide to Great Ideas

Praise for the First Edition «. . . an enchanting book for those people in computer science or mathematics who are fascinated by the concept of infinity.»—Computing Reviews «. . . a very well written introduction to set theory . . . easy to read and well suited for self-study . . . highly recommended.»—Choice The concept of infinity has fascinated and confused mankind for centuries with theories and ideas that cause even seasoned mathematicians to wonder. The Mathematics of Infinity: A Guide to Great Ideas, Second Edition uniquely explores how we can manipulate these ideas when our common sense rebels at the conclusions we are drawing. Continuing to draw from his extensive work on the subject, the author provides a user-friendly presentation that avoids unnecessary, in-depth mathematical rigor. This Second Edition provides important coverage of logic and sets, elements and predicates, cardinals as ordinals, and mathematical physics. Classic arguments and illustrative examples are provided throughout the book and are accompanied by a gradual progression of sophisticated notions designed to stun readers' intuitive view of the world. With an accessible and balanced treatment of both concepts and theory, the book focuses on the following topics: Logic, sets, and functions Prime numbers Counting infinite sets Well ordered sets Infinite cardinals Logic and meta-mathematics Inductions and numbers Presenting an intriguing account of the notions of infinity, The Mathematics of Infinity: A Guide to Great Ideas, Second Edition is an insightful supplement for mathematics courses on set theory at the undergraduate level. The book also serves as a fascinating reference for mathematically inclined individuals who are interested in learning about the world of counterintuitive mathematics.

7126.78 рублей

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Theodore Faticoni G. The Mathematics of Infinity. A Guide to Great Ideas

Praise for the First Edition «. . . an enchanting book for those people in computer science or mathematics who are fascinated by the concept of infinity.»—Computing Reviews «. . . a very well written introduction to set theory . . . easy to read and well suited for self-study . . . highly recommended.»—Choice The concept of infinity has fascinated and confused mankind for centuries with theories and ideas that cause even seasoned mathematicians to wonder. The Mathematics of Infinity: A Guide to Great Ideas, Second Edition uniquely explores how we can manipulate these ideas when our common sense rebels at the conclusions we are drawing. Continuing to draw from his extensive work on the subject, the author provides a user-friendly presentation that avoids unnecessary, in-depth mathematical rigor. This Second Edition provides important coverage of logic and sets, elements and predicates, cardinals as ordinals, and mathematical physics. Classic arguments and illustrative examples are provided throughout the book and are accompanied by a gradual progression of sophisticated notions designed to stun readers' intuitive view of the world. With an accessible and balanced treatment of both concepts and theory, the book focuses on the following topics: Logic, sets, and functions Prime numbers Counting infinite sets Well ordered sets Infinite cardinals Logic and meta-mathematics Inductions and numbers Presenting an intriguing account of the notions of infinity, The Mathematics of Infinity: A Guide to Great Ideas, Second Edition is an insightful supplement for mathematics courses on set theory at the undergraduate level. The book also serves as a fascinating reference for mathematically inclined individuals who are interested in learning about the world of counterintuitive mathematics.

6973.85 рублей

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V. Narayanamurthy Applied Impact Mechanics

This book is intended to help the reader understand impact phenomena as a focused application of diverse topics such as rigid body dynamics, structural dynamics, contact and continuum mechanics, shock and vibration, wave propagation and material modelling. It emphasizes the need for a proper assessment of sophisticated experimental/computational tools promoted widely in contemporary design. A unique feature of the book is its presentation of several examples and exercises to aid further understanding of the physics and mathematics of impact process from first principles, in a way that is simple to follow.

8997.79 рублей

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Nicolas Turenne Knowledge Needs and Information Extraction. Towards an Artificial Consciousness

This book presents a theory of consciousness which is unique and sustainable in nature, based on physiological and cognitive-linguistic principles controlled by a number of socio-psycho-economic factors. In order to anchor this theory, which draws upon various disciplines, the author presents a number of different theories, all of which have been abundantly studied by scientists from both a theoretical and experimental standpoint, including models of social organization, ego theories, theories of the motivational system in psychology, theories of the motivational system in neurosciences, language modeling and computational modeling of motivation. The theory presented in this book is based on the hypothesis that an individual’s main activities are developed by self-motivation, managed as an informational need. This is described in chapters covering self-motivation on a day-to-day basis, the notion of need, the hypothesis and control of cognitive self-motivation and a model of self-motivation which associates language and physiology. The subject of knowledge extraction is also covered, including the impact of self-motivation on written information, non-transversal and transversal text-mining techniques and the fields of interest of text mining. Contents: 1. Consciousness: an Ancient and Current Topic of Study. 2. Self-motivation on a Daily Basis. 3. The Notion of Need. 4. The Models of Social Organization. 5. Self Theories. 6. Theories of Motivation in Psychology. 7. Theories of Motivation in Neurosciences. 8. Language Modeling. 9. Computational Modeling of Motivation. 10. Hypothesis and Control of Cognitive Self-Motivation. 11. A Model of Self-Motivation which Associates Language and Physiology. 12. Impact of Self-Motivation on Written Information. 13. Non-Transversal Text Mining Techniques. 14. Transversal Text Mining Techniques. 15. Fields of Interest for Text Mining. About the Authors Nicolas Turenne is a researcher at INRA in the Science and Society team at the University of Paris-Est Marne la Vallée in France. He specializes in knowledge extraction from texts with theoretical research into relational and stochastic models. His research topics also concern the sociology of uses, food and environmental sciences, and bioinformatics.

8472.77 рублей

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Willem Conradie Logic and Discrete Mathematics. A Concise Introduction

A concise yet rigorous introduction to logic and discrete mathematics. This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade. The chapters on logic – propositional and first-order – provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language and the semantics of classical logic as well as practical applications through the easy to understand and use deductive systems of Semantic Tableaux and Resolution. The chapters on set theory, number theory, combinatorics and graph theory combine the necessary minimum of theory with numerous examples and selected applications. Written in a clear and reader-friendly style, each section ends with an extensive set of exercises, most of them provided with complete solutions which are available in the accompanying solutions manual. Key Features: Suitable for a variety of courses for students in both Mathematics and Computer Science. Extensive, in-depth coverage of classical logic, combined with a solid exposition of a selection of the most important fields of discrete mathematics Concise, clear and uncluttered presentation with numerous examples. Covers some applications including cryptographic systems, discrete probability and network algorithms. Logic and Discrete Mathematics: A Concise Introduction is aimed mainly at undergraduate courses for students in mathematics and computer science, but the book will also be a valuable resource for graduate modules and for self-study.

4498.89 рублей

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Guo Yu Vibro-impact Dynamics

Presents a systematic view of vibro-impact dynamics based on the nonlinear dynamics analysis Comprehensive understanding of any vibro-impact system is critically impeded by the lack of analytical tools viable for properly characterizing grazing bifurcation. The authors establish vibro-impact dynamics as a subset of the theory of discontinuous systems, thus enabling all vibro-impact systems to be explored and characterized for applications. Vibro-impact Dynamics presents an original theoretical way of analyzing the behavior of vibro-impact dynamics that can be extended to discontinuous dynamics. All topics are logically integrated to allow for vibro-impact dynamics, the central theme, to be presented. It provides a unified treatment on the topic with a sound theoretical base that is applicable to both continuous and discrete systems Vibro-impact Dynamics: Presents mapping dynamics to determine bifurcation and chaos in vibro-impact systems Offers two simple vibro-impact systems with comprehensive physical interpretation of complex motions Uses the theory for discontinuous dynamical systems on time-varying domains, to investigate the Fermi-oscillator Essential reading for graduate students, university professors, researchers and scientists in mechanical engineering.

12257.29 рублей

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Selva Lewin-Bizan Thriving in Childhood and Adolescence: The Role of Self Regulation Processes. New Directions for Child and Adolescent Development, Number 133

Opening with a discussion on the need to integrate self-regulation processes and to create a life-span oriented framework of these processes, this volume explores several perspectives in the current scholarship. Chapter contributors examine theoretical concepts including Vygotsky/Luria Insights in the Development of Executive Functions Self-Regulation and Academic Achievement in Elementary School Children Influences of Children?s and Adolescents? Action-Control Processes on School Achievement, Peer Relationships, and Coping with Challenging Life Events Intentional Self-Regulation, Ecological Assets, and Thriving in Adolescence: A Developmental Systems Model and a Life-Span, Relational, Public Health Model of Self- Regulation: Impact on Individual and Community Health The volume concludes with New Directions for Child and Adolescent Development series editor-in-chief Reed W. Larson discussing the challenges reported by youth working on arts, technology, and social justice projects in organized programs and how they learn to address them. This is the 133nd volume of the Jossey-Bass quarterly report series New Directions for Child and Adolescent Development. The mission of this series is to provide scientific and scholarly presentations on cutting edge issues and concepts in the field of child and adolescent development. Each volume focuses on a specific new direction or research topic, and is edited by an expert or experts on that topic.

2174.04 рублей

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Brian Evans The Development of Mathematics Throughout the Centuries. A Brief History in a Cultural Context

Throughout the book, readers take a journey throughout time and observe how people around the world have understood these patterns of quantity, structure, and dimension around them. The Development of Mathematics Throughout the Centuries: A Brief History in a Cultural Contex provides a brief overview of the history of mathematics in a very straightforward and understandable manner and also addresses major findings that influenced the development of mathematics as a coherent discipline. This book: Highlights the contributions made by various world cultures including African, Egyptian, Babylonian, Chinese, Indian, Islamic, and pre-Columbian American mathematics Features an approach that is not too rigorous and is ideal for a one-semester course of the history of mathematics. Includes a Resources and Recommended Reading section for further exploration and has been extensively classroom-tested

6973.85 рублей

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Valerio Lucarini Extremes and Recurrence in Dynamical Systems

Written by a team of international experts, Extremes and Recurrence in Dynamical Systems presents a unique point of view on the mathematical theory of extremes and on its applications in the natural and social sciences. Featuring an interdisciplinary approach to new concepts in pure and applied mathematical research, the book skillfully combines the areas of statistical mechanics, probability theory, measure theory, dynamical systems, statistical inference, geophysics, and software application. Emphasizing the statistical mechanical point of view, the book introduces robust theoretical embedding for the application of extreme value theory in dynamical systems. Extremes and Recurrence in Dynamical Systems also features: • A careful examination of how a dynamical system can serve as a generator of stochastic processes • Discussions on the applications of statistical inference in the theoretical and heuristic use of extremes • Several examples of analysis of extremes in a physical and geophysical context • A final summary of the main results presented along with a guide to future research projects • An appendix with software in Matlab® programming language to help readers to develop further understanding of the presented concepts Extremes and Recurrence in Dynamical Systems is ideal for academics and practitioners in pure and applied mathematics, probability theory, statistics, chaos, theoretical and applied dynamical systems, statistical mechanics, geophysical fluid dynamics, geosciences and complexity science. VALERIO LUCARINI, PhD, is Professor of Theoretical Meteorology at the University of Hamburg, Germany and Professor of Statistical Mechanics at the University of Reading, UK. DAVIDE FARANDA, PhD, is Researcher at the Laboratoire des science du climat et de l’environnement, IPSL, CEA Saclay, Université Paris-Saclay, Gif-sur-Yvette, France. ANA CRISTINA GOMES MONTEIRO MOREIRA DE FREITAS, PhD, is Assistant Professor in the Faculty of Economics at the University of Porto, Portugal. JORGE MIGUEL MILHAZES DE FREITAS, PhD, is Assistant Professor in the Department of Mathematics of the Faculty of Sciences at the University of Porto, Portugal. MARK HOLLAND, PhD, is Senior Lecturer in Applied Mathematics in the College of Engineering, Mathematics and Physical Sciences at the University of Exeter, UK. TOBIAS KUNA, PhD, is Associate Professor in the Department of Mathematics and Statistics at the University of Reading, UK. MATTHEW NICOL, PhD, is Professor of Mathematics at the University of Houston, USA. MIKE TODD, PhD, is Lecturer in the School of Mathematics and Statistics at the University of St. Andrews, Scotland. SANDRO VAIENTI, PhD, is Professor of Mathematics at the University of Toulon and Researcher at the Centre de Physique Théorique, France.

8247.97 рублей

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Katherine Sibley A.S. A Companion to First Ladies

This volume explores more than two centuries of literature on the First Ladies, from Martha Washington to Michelle Obama, providing the first historiographical overview of these important women in U.S. history. Underlines the growing scholarly appreciation of the First Ladies and the evolution of the position since the 18th century Explores the impact of these women not only on White House responsibilities, but on elections, presidential policies, social causes, and in shaping their husbands’ legacies Brings the First Ladies into crisp historiographical focus, assessing how these women and their contributions have been perceived both in popular literature and scholarly debate Provides concise biographical treatments for each First Lady

3370.61 рублей

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Mohan R. Lavi The Impact of IFRS on Industry

The industry-specific guide to IFRS interpretation and application One of the biggest challenges of any reporting standard is how best to interpret and implement it in the context of a specific company or industry, given that each industry has its own intricacies and nuances. The Impact of IFRS on Industry provides specific guidance on applying IFRS in a diverse range of sectors. Opening with an overview of IFRS, including a summary of all the standards, the book goes on to provide detailed coverage of the standards that can impact all industries, including IAS-40, IFRS 1, and IFRS 5. Thereafter, it offers practical advice and guidance on the application of IFRS in specific industries, including automotive; oil and gas; bio-sciences; infrastructure; airlines; media and communications; government-owned entities; mining; software; banking and financial services; insurance; FMCG; shipping; pharmaceuticals; telecoms; real estate and construction; power; SMEs; retail; e-commerce; and the service sector. The book concludes with a discussion on the collateral impact of implementing IFRS and how forthcoming IFRS Standards could impact specific industries. Worked examples are used throughout to demonstrate how the standards are applied in practice. Understand the IFRS standards comprehensively Learn which standards impact all industries Examine the ways in which IFRS is applied in practice See how different standards are applied in specific industries Suitable as a quick reference or a comprehensive guide, The Impact of IFRS on Industry gives you the real-world IFRS answers you need.

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Tahira Farooqui Diet and Exercise in Cognitive Function and Neurological Diseases

Diet and exercise have long been recognized as important components of a healthy lifestyle, as they have a great impact on improving cardiovascular and cerebrovascular functions, lowering the risk of metabolic disorders, and contributing to healthy aging. As a greater proportion of the world’s population is living longer, there has been increased interest in understanding the role of nutrition and exercise in long-term neurological health and cognitive function. Diet and Exercise in Cognitive Function and Neurological Diseases discusses the role and impact that nutrition and activity have on cognitive function and neurological health. The book is divided into two sections. The first section focuses on diet and its impact on neurobiological processes. Chapters focus on the impacts of specific diets, such as the Mediterranean, ketogenic and vegan diets, as well as the role of specific nutrients, fats, fatty acids, and calorie restriction on neurological health and cognitive function. The second section of the book focuses on exercise, and its role in maintaining cognitive function, reducing neuroinflammatory responses, regulating adult neurogenesis, and healthy brain aging. Other chapters look at the impact of exercise in the management of specific neurological disorders such Multiple Sclerosis and Parkinson’s Disease. Diet and Exercise in Cognitive Function and Neurological Diseases is a timely reference on the neurobiological interplay between diet and exercise on long-term brain health and cognitive function.

15443.07 рублей

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Jed Emerson The Impact Investor. Lessons in Leadership and Strategy for Collaborative Capitalism

Your money can change the world The Impact Investor: Lessons in Leadership and Strategy for Collaborative Capitalism offers precise details on what, exactly, impact investing entails, embodied in the experiences and best and proven practices of some of the world's most successful impact investors, across asset classes, geographies and areas of impact. The book discusses the parameters of impact investing in unprecedented detail and clarity, providing both context and tools to those eager to engage in the generational shift in the way finance and business is being approached in the new era of Collaborative Capitalism. The book presents a simple thesis with clarity and conviction: «Impact investing can be done successfully. This is what success looks like, and this is what it requires.» With much-needed lessons for practitioners, the authors view impact investing as a harbinger of a new, more «multilingual» (cross-sector), transparent, and accountable form of economic leadership. The Impact Investor: Lessons in Leadership and Strategy for Collaborative Capitalism serves as a resource for a variety of players in finance and business, including: Investors: It demonstrates not only the types of investments which can be profitable and impactful, but also details best practices that, with roots in impact investing, will increasingly play a role in undergirding the success of all investment strategies. Wealth advisors/financial services professionals: With unprecedented detail on the innovative structures and strategies of impact investing funds, the book provides guidance to financial institutions on how to incorporate these investments in client portfolios. Foundations: The book explores the many catalytic and innovative ways for for-profit and non-profit investors to partner, amplifying the potential social and environmental impacts of philanthropic spending and market-rate endowment investment. Business students: By including strategies for making sound impact investments based on detailed case studies, it provides concrete lessons and explores the skills required to enhance prospects for success as a finance and business professional. Policy makers: Reinforcing the urgency of creating a supportive and enabling environment for impact investing, the book demonstrates ways policy has already shaped the sector, and suggests new ways for policymakers to support it. Corporate leaders: The book includes essential advice on the way business is and must be responding to a new generation of Millennial clients and customers, with unique insights into a form of value creation that is inherently more collaborative and outcomes-driven.

2551.99 рублей

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Written by a team of international experts, Extremes and Recurrence in Dynamical Systems presents a unique point of view on the mathematical theory of extremes and on its applications in the natural and social sciences. Featuring an interdisciplinary approach to new concepts in pure and applied mathematical research, the book skillfully combines the areas of statistical mechanics, probability theory, measure theory, dynamical systems, statistical inference, geophysics, and software application. Emphasizing the statistical mechanical point of view, the book introduces robust theoretical embedding for the application of extreme value theory in dynamical systems. Extremes and Recurrence in Dynamical Systems also features: • A careful examination of how a dynamical system can serve as a generator of stochastic processes • Discussions on the applications of statistical inference in the theoretical and heuristic use of extremes • Several examples of analysis of extremes in a physical and geophysical context • A final summary of the main results presented along with a guide to future research projects • An appendix with software in Matlab® programming language to help readers to develop further understanding of the presented concepts Extremes and Recurrence in Dynamical Systems is ideal for academics and practitioners in pure and applied mathematics, probability theory, statistics, chaos, theoretical and applied dynamical systems, statistical mechanics, geophysical fluid dynamics, geosciences and complexity science. VALERIO LUCARINI, PhD, is Professor of Theoretical Meteorology at the University of Hamburg, Germany and Professor of Statistical Mechanics at the University of Reading, UK. DAVIDE FARANDA, PhD, is Researcher at the Laboratoire des science du climat et de l’environnement, IPSL, CEA Saclay, Université Paris-Saclay, Gif-sur-Yvette, France. ANA CRISTINA GOMES MONTEIRO MOREIRA DE FREITAS, PhD, is Assistant Professor in the Faculty of Economics at the University of Porto, Portugal. JORGE MIGUEL MILHAZES DE FREITAS, PhD, is Assistant Professor in the Department of Mathematics of the Faculty of Sciences at the University of Porto, Portugal. MARK HOLLAND, PhD, is Senior Lecturer in Applied Mathematics in the College of Engineering, Mathematics and Physical Sciences at the University of Exeter, UK. TOBIAS KUNA, PhD, is Associate Professor in the Department of Mathematics and Statistics at the University of Reading, UK. MATTHEW NICOL, PhD, is Professor of Mathematics at the University of Houston, USA. MIKE TODD, PhD, is Lecturer in the School of Mathematics and Statistics at the University of St. Andrews, Scotland. SANDRO VAIENTI, PhD, is Professor of Mathematics at the University of Toulon and Researcher at the Centre de Physique Théorique, France.

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